Do you know of any instance when a continued fraction is really of use? Are continued fractions just an abstract ruse, or a more natural way to describe a ratio? Is there any paractical use? when sucha fraction is really necessary?

Continued fractions provide another representation of real numbers, offering insights that are not revealed by the decimal representation. For example, the [golden ratio has the continued fraction [1; 1, 1, ...]]( and [_e_ = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, ...]]( Continued fractions can be used to—

* approximate real numbers and functions,
* compute square roots,
* solve Pell's equation,
* break RSA encryption,
* prove the sum-of-two-squares theorem,
* recognize rational numbers,
* find lattice points close to curves, which is useful in the study of dynamical systems, and
* estimate eigenvalues and eigenvectors.